Problem: $ D = \left[\begin{array}{rrr}-2 & -2 & 1 \\ 2 & -1 & 1 \\ 2 & -1 & 4\end{array}\right]$ $ E = \left[\begin{array}{r}1 \\ 2 \\ -1\end{array}\right]$ Is $ D- E$ defined?
In order for subtraction of two matrices to be defined, the matrices must have the same dimensions. If $ D$ is of dimension $( m \times  n)$ and $ E$ is of dimension $( p \times  q)$ , then for their difference to be defined: 1. $ m$ (number of rows in $ D$ ) must equal $ p$ (number of rows in $ E$ ) and 2. $ n$ (number of columns in $ D$ ) must equal $ q$ (number of columns in $ E$ Do $ D$ and $ E$ have the same number of rows? Yes Yes No Yes Do $ D$ and $ E$ have the same number of columns? No Yes No No Since $ D$ has different dimensions $(3\times3)$ from $ E$ $(3\times1)$, $ D- E$ is not defined.